Closed-Form Solution of the Bending Two-Phase Integral Model of Euler-Bernoulli Nanobeams

Abstract

Recent developments have shown that the widely used simplified differential model of Eringen’s nonlocal elasticity in nanobeam analysis is not equivalent to the corresponding and initially proposed integral models, the pure integral model and the two-phase integral model, in all cases of loading and boundary conditions. This has resolved a paradox with solutions that are not in line with the expected softening effect of the nonlocal theory that appears in all other cases. In addition, it revived interest in the integral model and the two-phase integral model, which were not used due to their complexity in solving the relevant integral and integro-differential equations, respectively. In this article, we use a direct operator method for solving boundary value problems for nth order linear Volterra–Fredholm integro-differential equations of convolution type to construct closed-form solutions to the two-phase integral model of Euler–Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions.

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

“…We propose in this article another factorization method on two factors which successfully was applied in the articles by the authors [6], [7], [21]- [26] and by another author in [27]. Here we generalize the results of these papers and study a more complicated boundary value problem with an abstract operator equation…”

“…Note that the operator B, by Theorem 3 (iii), since (24) and bijectivity of A, is bijective, and that taking into account (12) the system of equations ( 25), (26) can be represented as S 0 = B −1 V and G 0 = B −1 Y. The last system, because of bijectivity of B, is equivalent to the system V = BS 0 and Y = BG 0 , which is the system (7), (8). Then by Theorem 3 (i), the operator B 1 can be factorized in B 1 = BB 0 .…”

“…Integro-differential equations (IDEs) are used to model many problems in science, engineering, economics, medicine, control theory, micro-inhomogeneous media and viscoelasticity [1][2][3][4][5][6][7][8][9]. Very important tools in solving of Boundary Value Problems (BVPs) with IDEs are the Parametrization Method [10] and the Factorization (Decomposition) method, but the applicability of the last method is confined to certain kinds of integro-differential operators, corresponding to BVPs and cannot be universal for all problems.…”

“…on a Banach space X for solving boundary value problems for n-th order linear Volterra-Fredholm integro-differential equations of convolution type, was used in [6], [7], where were constructed the closed-form solutions to the two-phase integral model of Euler-Bernoulli nanobeams in bending under transverse distributed load and various types of boundary conditions. In [21] the operator B 1 corresponding to the abstract operator equation…”

An Algorithm for the Closed-Form Solution of Certain Classes of Volterra–Fredholm Integral Equations of Convolution Type

“…appears in the one-dimensional transport process [111] and the bending analysis of nonlocal integral models of Euler-Bernoulli nanobeams [112][113][114]. This equation, by removing the modulus in the integrand, can be converted to a Volterra-Fredholm integral equation of convolution type, namely…”